Density interfaces in atmosphere and ocean fluid dynamics move up and down for a variety of reasons, some are related to fast waves such as gravity waves and other to slower vortical motions describing broader currents and weather systems. An important question is how to disentangle these different dynamical mechanisms from limited data sets taken along one-dimensional transects by following a boat or an airplane. This talk will describe the major issues that arise and some recent breakthroughs in mathematical data analysis techniques developed specifically for this problem.

I will present a computational alternative to probabilistic simulations for path-dependent stochastic dynamical systems that are prevalent in engineering mechanics. By way of example, we target (a) stochastic elasto-plasticity (involving transition between elastic and plastic states) and (b) obstacle problems with noise (involving discrete impulses due to collisions with an obstacle). We focus on solving Backward Kolmogorov Equations (BKEs) originating from elasto-plastic and obstacle oscillators. The main challenge in solving BKEs corresponding to these problems is to deal with the non-standard boundary conditions which describe the behavior of the underlying process on the boundary. Applications that could make use of this framework abound in many areas of science and technology. This is a joint work with Georg Stadler and Jonathan Wylie.

Many mammals (cat, monkey) possess ordered maps of orientation preference in the primary visual cortex. On the other hand, rodents (rat, mouse) have orientation preference mapped in a disordered or "salt and pepper" fashion. We develop a large-scale computational model of the input layer of primary visual cortex for Macaque monkey and for mouse -- models that capture the effects on orientation selectivity of ordered vs disordered maps of orientation preference. The mouse model reproduces the effects on "thalamus to cortical", and "cortical to cortical", excitation which have been observed in recent optogenetic experiments from the Scanziani Lab. We analyze the mechanisms by which the computer model achieves these effects in the presence of a disordered map of orientation preference.

The threshold dynamics method developed by Merriman, Bence and Osher(MBO) is an efficient method for simulating the motion by mean curvature flow when the interface is away from the solid boundary. Direct generalization of the MBO type method to the wetting problems with interface intersecting the solid boundary is not easy because solving heat equation on general domain with wetting boundary condition is not as efficient as that for the original MBO method. The dynamics of the contact point also follows a different dynamic law compared to interface dynamics away from the boundary. We develop an efficient volume preserving threshold dynamics (MBO) method for drop spreading on rough surfaces. The method is based on minimization of the weighted surface area functional over a extended domain that includes the solid phase. The method is simple, stable with the complexity O (N log N) per time step and it is not sensitive to the inhomogeneity or roughness of the solid boundary. We also extend the idea to an efficient method for image segmentation.

We consider the mathematical relation between diffuse interface and sharp interface models for the flow of two viscous, incompressible Newtonian fluids like oil and water. In diffuse interface models a partial mixing of the macroscopically immiscible fluids on a small length scale ε > 0 is taken into account. These models are capable to describe such two-phase flows beyond the occurrence of topological singularities of the interface due to collision or droplet formation. Both for theoretical and numerical purposes a deeper understanding of the limit ε → 0 is of interest. We discuss several rigorous mathematical results on convergence of diffuse interface to sharp interface models in dependence of the scaling.

For suitable initial data, we construct infinitely many weak solutions to the nematic liquid crystal flows in dimension three. The solutions are in the axisymmetric class with ''backward bubbling'' at a large time and bounded energy for any finite time.

The formation and dynamics of interfaces are often dictated by an underlying energy functional and the medium they are located in. The variational energy functional can either be from the interface or the bulk or from a mixture of both.In the former case, the geometric aspect of interfaces may often be related to "minimal surfaces", the latter two can vary in various situations. In this talk, I shall discuss the optimal partition problem for Dirichlet eigenvalues. Such problems appeared in classical optimal designs, and recently also in Data Searchings. It is a typical class of problem in which the interfaces are driven by the bulk energy. I shall discuss its existence, regularity and the problem of the large N asymptotics.